Contents Preface i 1.Introduction 1 1.1 The weighted residual methods 3 1.1.1 Problem description 3 1.1.2 Primal methods 4 1.1.3 Mixed methods 8 1.2 Application of weighted residual methods 9 1.2.1 Transient motions 10 1.2.2 Periodic motions 12 1.3 Finite difference methods 14 1.3.1 Explicit methods 15 1.3.2 Implicit methods 16 1.4 Asymptotic methods 16 1.4.1 Perturbation method 16 1.4.2 Adomian decomposition method 19 1.4.3 Picard iteration method 22 References 22 2.Harmonic Balance Method and Time Domain Collocation Method 27 2.1 Time collocation in a period of oscillation 29 2.2 Relationship between collocation and harmonic balance 31 2.2.1 Harmonic balance method 31 2.2.2 High dimensional harmonic balance method 33 2.2.3 Equivalence between HDHB and collocation 35 2.3 Initialization of Newton-Raphson method 38 2.3.1 Initial values for undamped system 39 2.3.2 Initial values for damped system 41 2.4 Numerical examples 42 2.4.1 Undamped Duffing equation42 2.4.2 Damped Duffing equation 47 Appendix A 51 Appendix B 51 References 53 3.Dealiasing for Harmonic Balance and Time Domain Collocation Methods 55 3.1 Governing equations of the airfoil model 56 3.2 Formulation of the HB method 59 3.2.1 Numerical approximation of Jacobian matrix 61 3.2.2 Explicit Jacobian matrix of HB 62 3.2.3 Mathematical aliasing of HB method 67 3.3 Formulation of the TDC method 68 3.3.1 Explicit Jacobian matrix of TDC 72 3.3.2 Mathematical aliasing of the TDC method 73 3.4 Reconstruction harmonic balance method 79 3.5 Numerical examples 80 3.5.1 RK4 results and spectral analysis 80 3.5.2 HBEJ vs.HBNJ 84 3.5.3 Aliasing analysis of the HB and TDC methods 87 3.5.4 Dealiasing via a marching procedure 93 Appendix 96 References 101 4.Application of Time Domain Collocation in FormationFlying of Satellites 103 4.1 TDC searching scheme for periodic relative orbits 104 4.2 Initial values for TDC method 109 4.2.1 The C-W equations 110 4.2.2 The T-H equations 112 4.3 Evaluation of TDC search scheme 112 4.3.1 Projected closed orbit 112 4.3.2 Closed loop control 113 4.4 Numerical results 114 Appendix 123 References 125 5.Local Variational Iteration Method 127 5.1 VIM and its relationship with PIM and ADM 130 5.1.1 VIM 130 5.1.2 Comparison of VIM with PIM and ADM 132 5.2 Local variational iteration method134 5.2.1 Limitations of global VIM 134 5.2.2 Variational homotopy method138 5.2.3 Methodology of LVIM 140 5.3 Conclusion 145 References 145 6.Collocation in Conjunction with the Local Variational Iteration Method 147 6.1 Modifications of LVIM 149 6.1.1 Algorithm-1 150 6.1.2 Algorithm-2 152 6.1.3 Algorithm-3 154 6.2 Implementation of LVIM156 6.2.1 Discretization using collocation 156 6.2.2 Collocation of algorithm-1157 6.2.3 Collocation of algorithm-2157 6.2.4 Collocation of algorithm-3159 6.3 Numerical examples 160 6.3.1 The forced Duffing equation 162 6.3.2 The Lorenz system 165 6.3.3 The multiple coupled Duffing equations 168 6.4 Conclusion 172 References 173 7.Application of the Local VariationalIteration Method in Orbital Mechanics 175 7.1 Local variational iteration method and quasi-linearization method 176 7.1.1 Local variational iteration method176 7.1.2 Quasi-linearization method178 7.2 Perturbed orbit propagation 181 7.2.1 Comparison of local variational iteration method with the modified Chebyshev picard iteration method 182 7.2.2 Comparison of FAPI with Runge-Kutta 12(10) 185 7.3 Perturbed Lambert's problem 187 7.3.1 Using FAPI 188 7.3.2 Using the fish-scale-growing method 189 7.3.3 Using quasilinearization and local variational iteration method 193 7.4 Conclusion 196 References 196 8.Applications of the Local Variational Iteration Method in Structural Dynamics 199 8.1 Elucidation of LVIM in structural dynamics 200 8.1.1 Formulas of the local variational iteration method 200 8.1.2 Large time interval collocation 202 8.1.3 LVIM algorithms for structural dynamical system 203 8.2 Mathematical model of a buckled beam 208 8.3 Nonlinear vibrations of a buckled beam 211 8.3.1 Bifurcations and chaos 211 8.3.2 Comparison between HHT and LVIM algorithms 218 8.4 Conclusion 223 Appendix A 224 Appendix B 224 Appendix C 225 References 225 Index 227
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